A216222 -id:A216222 - cp's OEIS frontend

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 4, 2, 3, 3, 2, 6, 2, 7, 2, 8, 3, 10, 6, 8, 9, 8, 12, 8, 16, 6, 20, 8, 22, 10, 24, 14, 27, 20, 26, 26, 25, 34, 26, 42, 25, 51, 26, 58, 31, 66, 36, 72, 43, 76, 56, 82, 70, 82, 86, 84, 106, 87, 124, 90, 145, 95, 168, 102, 187, 115, 206

Offset: 0

Vaclav Kotesovec, Sep 28 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A216222, A306734, A333179, A340647, A369557, A376530.

nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ A369557(n) / 4.

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 3, 3, 3, 4, 4, 5, 5, 7, 9, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 19, 24, 23, 25, 27, 28, 31, 32, 33, 37, 40, 42, 44, 47, 52, 54, 59, 62, 67, 75, 75, 80, 87, 90, 95, 102, 109, 114, 119, 127, 134, 142, 150, 159, 171, 178, 187, 199, 211

Offset: 0

Vaclav Kotesovec, Sep 29 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A053258, A216222, A306734, A369557, A376542, A376581, A376621.

nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.390989767113799449629...
a(n) ~ c * A376542(n), where c = (108 + 12*sqrt(93))^(1/3)/3 - 4/(108 + 12*sqrt(93))^(1/3) = 1.364655607... is the real root of the equation c*(4 + c^2) = 8.
a(n) ~ c * A369557(n), where c = A347178 = -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3) / sqrt(3) = 0.3411639019... is the real root of the equation 2*c*(1 + 4*c^2) = 1.
a(n) ~ A376631(n) * (A376621/A376660)^sqrt(n).

A376812 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^j)^2.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 5, 5, 5, 7, 8, 10, 13, 14, 16, 19, 21, 25, 29, 33, 40, 45, 50, 57, 64, 72, 81, 93, 104, 117, 134, 148, 165, 185, 204, 227, 253, 280, 310, 345, 381, 422, 469, 514, 567, 625, 685, 753, 825, 903, 990, 1086, 1186, 1297, 1419, 1548, 1692, 1845, 2007

Offset: 0

Vaclav Kotesovec, Oct 05 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A053261, A376626, A376627, A376813, A216222.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1)/2)*Product[1+x^k, {k, 1, n}]^2, {n, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (1 + x^j)^2 * x^j.

a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 1/(4*sqrt(3/2 - 2*sinh(arcsinh(3^(3/2)/2)/3)/sqrt(3))) = 0.27647151570071656262813536...

A376813 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} (1 + x^j)^2.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 2, 2, 3, 6, 7, 8, 10, 8, 8, 8, 6, 8, 10, 12, 16, 20, 22, 24, 27, 26, 25, 26, 25, 26, 29, 32, 37, 44, 52, 58, 66, 72, 76, 82, 82, 84, 87, 88, 91, 96, 103, 112, 126, 138, 154, 174, 190, 208, 225, 238, 253, 268, 275, 284, 296, 304

Offset: 0

Vaclav Kotesovec, Oct 05 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A333179, A376623, A376812, A216222.

nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[1+x^k, {k, 1, n}]^2, {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(2*j))^2.

a(n) ~ phi^(1/2) * exp(Pi*sqrt(2*n/15)) / (4 * 5^(1/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A376854 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

Original entry on oeis.org

1, 1, 4, 8, 13, 20, 32, 52, 84, 133, 204, 304, 444, 636, 900, 1264, 1761, 2440, 3364, 4608, 6276, 8496, 11424, 15268, 20284, 26789, 35196, 46016, 59884, 77612, 100204, 128900, 165260, 211200, 269072, 341792, 432917, 546788, 688728, 865200, 1084048, 1354816, 1689048

Offset: 0

Vaclav Kotesovec, Oct 06 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A066447, A000041, A216222, A376852, A376853.

nmax = 60; CoefficientList[Series[Sum[x^(k^2) * Product[(1+x^j)/(1-x^j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ (1 + sqrt(2)) * exp(Pi*sqrt(n)) / (2^(9/2) * n).

cp's OEIS Frontend

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

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A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

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A376812 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^j)^2.

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A376813 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)^2.

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A376854 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} ((1 + x^j)/(1 - x^j))^2.

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A376812 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^j)^2.

A376813 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} (1 + x^j)^2.